HOLDER CONTINUITY UP TO THE BOUNDARY OF ¨SOLUTIONS TO NONLINEAR FOURTH–ORDER ELLIPTIC EQUATIONS WITH NATURAL GROWTH TERMS

In a bounded open set Ω ⊂ Rn, n > 3, we consider the nonlinear fourth-order partial differential equation ∑|α|=1,2(−1)^|α|D^α A_α(x,u,Du,D^2 u)+B(x,u,Du,D^2 u) = 0. It is assumed that the principal coefficients {A_α}|α|=1,2 satisfy the growth and coercivity conditions suitablefor the energy space W˚_(1,q)^(2,p)(Ω) = W˚_(1,q)(Ω)∩W˚^(2,p)(Ω), 1 < p < n/2, 2p < q < n. The lower-order term B(x,u,Du,D^2 u) behaves as b(u)(|Du|^q +|D^2 u|^p) +g(x) where g ∈ L^τ(Ω), τ > n/q. Weestablish the Holder continuity up to the boundary of any solution u ∈W˚_(1,q)^(2,p)(Ω)∩L^∞(Ω) by using the measure density condition on ∂Ω, an interior local result and a modified Moser method with special test function.